Well-posedness of the continuity equation in the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ for vector fields beyond the Sobolev, BV, and repulsive-Hamiltonian classes

The almost-everywhere well-posedness theory for continuity equations with measure initial data developed by Ambrosio and Figalli in 2010 rests on a standing hypothesis: that the associated scalar continuity equation has existence and uniqueness for nonnegative densities in $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$. At the time of the Note, this hypothesis was verified essentially only for the classical DiPerna–Lions Sobolev and Ambrosio BV renormalization classes. The follow-up paper by Ambrosio, Figalli, Friesecke, Giannoulis, and Paul (Comm. Pure Appl. Math. 2011, with a 2013 corrigendum) established it for a class of Hamiltonian vector fields with repulsive singular potentials, motivated by semiclassical limits of the Schrödinger equation. The general question, left standing as an assumption in these works, is: identify sharp, verifiable conditions on the vector field $b$ that guarantee existence and uniqueness in this function class beyond the three known classes, and delimit the boundary by non-uniqueness counterexamples. Convex-integration non-uniqueness results for Sobolev fields (2018) and 2024–2026 typicality and delimitation results show this boundary is genuinely subtle and under active investigation; no work provides a characterization in the general time-dependent, arbitrary-dimension setting (a Sard-type characterization exists only for autonomous divergence-free planar fields).

Background

The continuity equation $\partial_t\mu_t+\nabla\cdot(b_t\mu_t)=0$ on $[0,T]\times\mathbb{R}^d$ describes the transport of a mass distribution $\mu_t$ (a positive measure, in general singular with respect to Lebesgue) by a possibly nonsmooth vector field $b$; it governs, for instance, the evolution of densities in fluid mechanics, kinetic theory, and the semiclassical limit of quantum mechanics (the limit of quantum dynamics as Planck's constant tends to zero). When the initial datum is a measure rather than an $L^1$ density, well-posedness cannot be reduced to classical characteristics, since $b$ may be too rough for the associated ODE to have a unique flow.

The foundational result for nonsmooth fields is the renormalization theory of DiPerna and Lions (1989) — renormalization means testing the equation against nonlinear functions $\beta(w)$ of the solution, using the chain rule in the weak sense, which is what Sobolev and $BV$ regularity of the field provides. If $b$ is Sobolev (that is, $b\in L^1_t W^{1,p}_x$ with $p>1$) and satisfies the linear growth bound $|b_t(x)|\leq C(1+|x|)$ for a.e. $(t,x)$, the transport/continuity equation is well-posed for $L^p$ densities. Ambrosio (2004) extended this to $BV$ vector fields (fields whose distributional derivative is a finite measure), introducing the regular Lagrangian flow, the flow obtained by pushing Lebesgue measure and characterized up to null sets. These are the two classical sufficient conditions on $b$.

Ambrosio and Figalli (C. R. Math. Acad. Sci. Paris 348 (2010) 249–252) studied a different question: measure-valued initial data. Their theorems give existence, uniqueness, and stability for the induced flow on the space of measures, valid for almost every initial datum with respect to a reference measure $\nu$ — hence "almost everywhere" well-posedness. All of these statements are proved under a standing hypothesis: the scalar equation $\partial_t w_t+\nabla\cdot(b_t w_t)=0$, $w_0=\bar w$, for nonnegative densities must have existence and/or uniqueness in the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ (densities uniformly bounded in time both in $L^1$ and $L^\infty$). This is exactly the regularity class produced by the Wigner and Husimi transforms (phase-space representations of quantum states used to pass from Schrödinger equations to classical dynamics) in the applications the authors had in mind. The Note recalls only the Sobolev/BV conditions as known verifiable sufficient conditions, and defers all proofs to a longer announced work.

That longer work appeared as Ambrosio, Figalli, Friesecke, Giannoulis, and Paul, Comm. Pure Appl. Math. 64 (2011) 1199–1242 (corrigendum: CPAM 66 (2013) 646–647). Beyond supplying the proofs, its Section 6 verifies the standing hypothesis for a genuinely new class: Hamiltonian fields $b=(p,-\nabla U(x))$ in phase space $\mathbb{R}^{2n}$ where $U$ is locally Lipschitz off a closed Lebesgue-negligible set $S$ with $\nabla U\in BV_{\mathrm{loc}}$ off $S$, $U(x)\to+\infty$ as $x\to S$ (a repulsive singularity, as for Coulomb-type potentials), and $|\nabla U|/(1+|x|)$ uniformly bounded on energy sublevels. The proof exploits conservation of the energy $|p|^2/2+U(x)$ to truncate the field into a $BV$ field. This covers the semiclassical applications but leaves general, non-Hamiltonian and attractive-singular fields untouched.

On the negative side, convex integration (a technique for constructing wildly oscillatory solutions by adding high-frequency perturbations, pioneered by De Lellis and Székelyhidi for fluid equations) yields strong non-uniqueness: Modena and Székelyhidi (Annals of PDE 4 (2018)) built divergence-free Sobolev $W^{1,p}$ fields in integrability regimes beyond the DiPerna–Lions condition $1/p+1/\tilde p\leq 1$ (their construction covers $1/p+1/\tilde p>1+1/(d-1)$, where $p$ is the integrability exponent of the density and $\tilde p$ that of the field) for which the transport equation has non-unique solutions, showing the Sobolev scale cannot be naively relaxed. Recent work of Cianfrocca and Modena (2025) shows the uniqueness/non-uniqueness boundary remains unsettled even in the scaling-invariant $L^1_t L^q_x$ classes: in certain regimes existence of solutions is a Baire-meager property while uniqueness is generic, yet Sobolev-type non-uniqueness examples are dense, and work on this boundary has continued through 2026. A 2017 survey by Ambrosio confirms that many sufficient classes are known but no characterization is available in the general time-dependent setting; the closest results are tied to special structures — Alberti, Bianchini, and Crippa (2014) characterize, via a Sard-type property of the potential of the field, the autonomous divergence-free fields on the plane for which the Cauchy problem admits a unique bounded weak solution for every initial datum, and Clop, Jylhä, Mateu, and Orobitg (2019) derive well-posedness from assumptions only on the growth and the modulus of continuity of the field — axes that do not cover the rough or singular fields at issue here. Other adjacent lines — well-posedness in metric measure spaces (Ambrosio–Trevisan 2014), uniqueness for measure classes that need not be absolutely continuous (Bogachev, Da Prato, Röckner, Shaposhnikov 2015), and the semiclassical applications literature (Figalli–Ligabó–Paul 2012; Fermanian Kammerer–Gérard–Lasser 2013; Chabu 2017; Athanassoulis–Paul 2013; Golse–Paul 2021; Lafleche 2021) — extend the ambient setting or the applications, not the conditions on $b$.

Problem Statement

Let $T>0$, $d\geq 1$, and let $b:[0,T]\times\mathbb{R}^d\to\mathbb{R}^d$ be a Borel vector field satisfying the standard linear-growth bound $|b_t(x)|/(1+|x|)\in L^\infty([0,T]\times\mathbb{R}^d)$ (the linear-growth ratio is essentially bounded in time and space). Consider the scalar continuity equation $\partial_t w_t+\nabla\cdot(b_t w_t)=0,\qquad w_0=\bar w\geq 0,$ for nonnegative densities. This is the standing hypothesis of the Ambrosio–Figalli almost-everywhere well-posedness theory for measure initial data: identify sharp, verifiable conditions on $b$ under which this equation has existence and uniqueness in the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ (nonnegative solutions uniformly bounded in time in $L^1$ and in $L^\infty$).

The question concerns classes of vector fields going beyond the three known ones: (a) Sobolev fields $b\in L^1_t W^{1,p}_x$, $p>1$ (DiPerna–Lions renormalization), (b) $BV$ fields (Ambrosio's regular Lagrangian flow theory), and (c) the repulsive-Hamiltonian singular class of Ambrosio–Figalli–Friesecke–Giannoulis–Paul (2011), where $b=(p,-\nabla U)$ with $U$ locally Lipschitz and $\nabla U\in BV_{\mathrm{loc}}$ off a negligible closed singular set at which $U\to+\infty$. The target has two jointly supporting components: (i) sufficient conditions on $b$, covering regimes not reducible to (a)–(c) — for example singular fields that are non-Hamiltonian, have attractive rather than repulsive singularities, or lack gradient structure — that provably yield existence and uniqueness in the stated class; and (ii) the complementary delimitation: constructions of vector fields, again beyond (a)–(c), for which uniqueness in the stated class fails. Each component is an independently refutable sub-question; the two components are complementary, and the verifiable conditions are sharp when matched against the counterexamples, so that the boundary between well-posedness and non-uniqueness for the class $L^\infty_+([0,T];L^1\cap L^\infty(\mathbb{R}^d))$ is characterized.

References

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